As mentioned, in the absence of random effects, Condition 1\(^{*}\)  and 2\(^{*}\)  will reduce to Condition 1 and 2 under the linear model. Conditions 1\(^{*}\)  and 2\(^{*}\)  can be viewed as extensions of Conditions 1 and 2 for longitudinal data. However, unlike the classic envelope condition [1]}, Condition 1\(^{*}\)  only requires the expectation of \({\mathbf {\Gamma }}_0^T\mathbf {Y}_i\mid \mathbf {X}_i, \mathbf {Z}_i\)  and \( {\mathbf {\Gamma }}_0^T\mathbf {Y}_i\)  to be the same. The motivation of Condition 1\(^{*}\)  is that instead of imposing the redundancy of \({\mathbf {\Gamma }}_0^T\mathbf {Y}_i\)  on its entire distribution, we just assume the redundancy on its mean, which is easier to satisfy.
